To put it simply your loads are causing the beam/bar to bend right? So at the cut we need to consider the internal bending moment ($M$).
Like you said: the professor has cut the bar to calculate the INTERNAL forces experienced by the member. In any 2D problem (say in the x-y plane) the first thing that you learn is that whenever you take a cut there's three internal forces you need to consider:
Shear Force ($V$ or $Q$) perpendicular to the longitudinal axis of the member.
Axial Force ($N$) along the longitudinal axis of the member.
Bending Moment ($M$) about the z-axis which is coming out of the plane/page.
We consider these three internal loads because we know that they can be experienced by the member along its length due to the reactions at the supports and the external loads. The one you seem confused about is the moment ($M$). Nearly all the time in statics your loads on the beam will cause bending and this why we need to add a force vector for bending moment whenever we take a cut. Imagine taking a ruler, holding it at both ends and causing it to bend in the middle - a bending force is there that needs to be considered.
Now just because we've taken a cut and considered that these three internal loads, it doesn't mean they ALWAYS exist. They MIGHT exist. For instance, if you take a cut of the beam/bar where shear force is 0, then bending moment is typically at its maximum. At the point where no internal shear force exists (i.e. it is switching direction), bending moment happens to be maximum. Another example is at free ends the bending moment is typically zero.